One of the new topics in both tiers (for all boards) is “use inequality notation to specify simple error intervals due to truncation or rounding”. I don’t get how using the notation is that useful, but the programme of study (POS) does include “use and interpret limits of accuracy” and then note the bit about “including upper and lower bounds” (UB and LB) which means that these remain at higher tier only. The following is not, in any way, shape or form meant to be the full extent of this topic …
As a topic, “error intervals with inequality notation” isn’t something I’ve specifically taught … I didn’t really get my head around it until recently so I’ll explain in more detail before giving you some of my random thoughts.
Upper bound – the largest value that a number can be (to a specified degree of accuracy)
Lower bound – the smallest value that a number can be (to a specified degree of accuracy)
Error interval – the range of values (between the upper and lower bounds) in which the precise value could be.
The maths bit:
Let me explain with some examples (below) . Remember for any number correct to a specified degree (unit) of accuracy, the exact value lies in a range from half a unit above to half a unit below the given number.
Some of the SAM’s give an insight into how different boards are examining this element of the programme of study – some are including questions explicitly asking for error intervals, and others are wrapping it up into context style questions and those requiring an explanation about how any assumptions they’ve made in their working affect the final answer. OCR gives the best examples of how it’s being examined as a “standalone” topic:
What was interesting is that OCR was the only board that explicitly included a question about “truncation” … it caught me out! This is slightly different in that the UB and LB are NOT +/- half a unit
However, I much prefer the style of question where limits of accuracy are given for say books on a shelf or kitchen cupboards and students have to justify their answers. I know this kind of thing is much more difficult to teach and for borderline students a formulaic approach doesn’t work as the context changes with each question, but from a “teaching for understanding” point of view these kind of questions tick all the boxes for me.
The fact that the “process” in deriving these error intervals is so wrapped up with upper and lower bounds, yet interpreting limits of accuracy including UB and LB are topics limited to the higher tier suggests that whoever is responsible for the programme of study hasn’t got a “Scooby-doo” or just didn’t think it through. It appears to be just another example of how the fact that maths is so interrelated has been ignored and feels like it goes against the principle of not teaching topics in isolated silos or maybe it’s a matter of how the POS is interpreted … I’m trying to play devil’s advocate for a change.
I for one will be introducing the terminology of UB/LB’s … I’m not sure how else we could justify “why” we are using certain values (in any subsequent calculations) without using the terms UB/LB but when it comes to “truncated” values I don’t think I’ll refer to the values as UB or LB’s as that’ll just cause confusion.
As a topic it could be reinforced when teaching most topics that ask for answers to a given degree of accuracy so something that I think should be introduced quite early on in high school maths.
Hopefully the above is of some use in relation to the new topics introduced in the new Big Fat GCSE … You may find this post about the product rule useful too.